A multiplicative ergodic theorem for lipschitz maps
If (Fn, n [greater-or-equal, slanted] 0) is a stationary (ergodic) sequence of Lipschitz maps of a locally compact Polish space X into itself having a.s. negative Lyapunov exponent function, the composition process Fn...F1x converges in distribution to a stationary (ergodic) process in X (independent of x). For every x, the empirical distribution of a trajectory converges with probability one, and for every [var epsilon]>0, almost every trajectory is eventually within [var epsilon] of the support. We use the fact that the Lyapunov exponent of a process "run backwards" is the same as forwards. A set invariance condition is given for the case when (Fn) is a Markov chain. The result has applications to computer graphics and stability in control theory.
Volume (Year): 34 (1990)
Issue (Month): 1 (February)
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